1.4 Trade Math II: Fractions, Offsets, Areas & Heat Calculations

Why this section matters

The second half of the Trade Math blueprint covers piping geometry and quick conversions. These are the skills a journeyman uses to cut a 45-degree offset, compare pipe capacities, or check a water heater's recovery. Every method below works with a four-function calculator. Where a square root is needed, the numbers are chosen to come out even, or you can estimate.

Fractions, decimals and feet

Rulers read in sixteenths, but calculators work in decimals.

• Decimal inches to sixteenths: multiply the decimal part by 16 and round. For example, $0.45 \times 16 = 7.2$, which rounds to 7/16.
• Inches to decimal feet: divide by 12. So 8 inches is 0.67 foot, and 15 inches is 1.25 feet.
• Percent grade: divide the slope in inches per foot by 12, then multiply by 100. That gives about 2.08% for 1/4 inch per foot, 1.04% for 1/8 and 0.52% for 1/16.

Areas, circumference and capacity ratios

$$A = 0.7854 \times d^2 \qquad C = 3.1416 \times d$$

A pipe's flow area grows with the square of its diameter:

• A 4-inch pipe has $(4 \div 2)^2 = 4$ times the area of a 2-inch pipe.
• A 3-inch pipe has $(3 \div 1.5)^2 = 4$ times the area of a 1-1/2-inch pipe.
• It takes four 2-inch pipes to equal the area of one 4-inch pipe, not two.

Circumference shows up in storm drainage too. IPC 1106.5 and 1108.3 require a scupper to be at least as wide as the circumference of the roof drain it replaces. For a 4-inch drain, that is $3.1416 \times 4 \approx 12.6$ inches (see section 8.2).

Water velocity in a pipe: $v\ (\text{ft/s}) \approx 0.4085 \times Q\ (\text{gpm}) \div d^2$, with $d$ the inside diameter in inches. For example, 20 gpm through a 1-inch pipe moves at about 8.2 feet per second.

Simple offsets

An offset moves a pipe line sideways and returns it to a parallel path with two equal fittings. Three measurements describe it:

• Set (offset): the distance between the centerlines of the two parallel runs.
• Travel: the center-to-center length of the angled piece.
• Run (advance): the distance gained along the original direction while making the offset.

$$\text{Travel} = \text{Set} \times \text{travel multiplier} \qquad \text{Run} = \text{Set} \times \text{run multiplier}$$

These multipliers are trigonometry. The travel multiplier is $1 \div \sin(\text{angle})$ and the run multiplier is $1 \div \tan(\text{angle})$.

Example. A 45-degree offset needs an 18-inch set. Travel is $18 \times 1.414 = 25.45$ inches. Since $0.45 \times 16 = 7.2$, that is about 25-7/16 inches center to center. Run equals set, 18 inches.

From center-to-center to cut length

Fittings take up length. The take-off of a fitting is its center-to-end dimension minus the depth the pipe goes into the socket. Use the manufacturer's dimension sheet, because take-offs vary by material and pattern.

$$\text{Cut length} = \text{Center-to-center} - \text{take-off of fitting 1} - \text{take-off of fitting 2}$$

Measurement terms used on drawings and in the math text:

• Center to center: between fitting centerlines.
• End to end: the overall pipe length.
• End to center: from a pipe end to the centerline of a fitting.
• Face to face: between the faces of fittings or flanges.

Rolling offsets

A rolling offset moves the pipe both sideways (roll) and up or down (rise) at once. Solve it in two steps:

1. True offset $= \sqrt{\text{roll}^2 + \text{rise}^2}$ (the Pythagorean theorem).
2. Travel $=$ true offset × the multiplier for the fitting angle.

Example. Roll 9 inches, rise 12 inches. True offset $= \sqrt{81 + 144} = \sqrt{225} = 15$ inches. With 45-degree fittings, travel $= 15 \times 1.414 = 21.2$ inches.

Remember the 3-4-5 right triangle and its multiples (6-8-10, 9-12-15, 12-16-20). Plumbers use it to square a layout without a calculator.

Temperatures and heat

$$°F = (°C \times 1.8) + 32 \qquad °C = (°F - 32) \div 1.8$$

Useful pairs: 49°C = 120°F (the IPC 412.3 shower limit), 43°C = 110°F (the 607.1.2 tempered water limit), 60°C = 140°F, and 99°C = 210°F (the 504.5 relief temperature).

Btu formula. One Btu raises 1 pound of water by 1°F.

$$\text{Btu} = \text{gallons} \times 8.34 \times \Delta T$$

Recovery example. A gas water heater has a 40,000 Btu/h input at 76% efficiency, so its output is $40{,}000 \times 0.76 = 30{,}400$ Btu/h. Raising incoming 50°F water to 140°F is a 90°F rise. Recovery is $30{,}400 \div (8.34 \times 90) = 30{,}400 \div 750.6 \approx 40.5$ gallons per hour.

Other conversions

Exam traps

• Wrong multiplier: 1.414 is for 45° fittings. A 60° offset uses 1.155, and 22-1/2° uses 2.613.
• Adding roll and rise: the true offset of a rolling offset is the square root of the sum of the squares, not the sum.
• Diameter vs. area: doubling the diameter quadruples the area.
• Btu problems: use the heater's output (input × efficiency), not its nameplate input, for recovery.